<html>
  <head>
    <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
    <title>armax1</title>
  </head>
  <body bgcolor="#FFFFFF">
    <center>Scilab Function</center>
    <div align="right">Last update : 16/12/2004</div>
    <p>
      <b>armax1</b> -  armax identification</p>
    <h3>
      <font color="blue">Calling Sequence</font>
    </h3>
    <dl>
      <dd>
        <tt>[arc,resid]=armax1(r,s,q,y,u [,b0f])  </tt>
      </dd>
    </dl>
    <h3>
      <font color="blue">Parameters</font>
    </h3>
    <ul>
      <li>
        <tt>
          <b>y</b>
        </tt>: output signal</li>
      <li>
        <tt>
          <b>u</b>
        </tt>: input signal</li>
      <li>
        <tt>
          <b>r,s,q</b>
        </tt>: auto regression orders with r &gt;=0, s &gt;=-1.</li>
      <li>
        <tt>
          <b>b0f</b>
        </tt>: optional parameter. Its default value is 0 and it means that the coefficient b0 must be identified. if bof=1 the b0 is supposed to be zero and is not identified</li>
      <li>
        <tt>
          <b>arc</b>
        </tt>: is tlist with type "ar" and fields a,  b,  d,  ny,  nu,  sig<ul>
          <li>
            <tt>
              <b>a</b>
            </tt>: is the vector <tt>
              <b>[1,a1,...,a_r]</b>
            </tt>
          </li>
          <li>
            <tt>
              <b>b</b>
            </tt>: is the vector <tt>
              <b>[b0,......,b_s]</b>
            </tt>
          </li>
          <li>
            <tt>
              <b>d</b>
            </tt>: is the vector <tt>
              <b>[1,d1,....,d_q]</b>
            </tt>
          </li>
          <li>
            <tt>
              <b>sig</b>
            </tt>:   resid=[ sig*echap(1),....,];</li>
        </ul>
      </li>
    </ul>
    <h3>
      <font color="blue">Description</font>
    </h3>
    <p>
    armax1 is used to identify the coefficients of a 1-dimensional 
    ARX process:</p>
    <pre>

   A(z^-1)y= B(z^-1)u + D(z^-1)sig*e(t)
   e(t) is a 1-dimensional white noise with variance 1.
   A(z)= 1+a1*z+...+a_r*z^r; ( r=0 =&gt; A(z)=1)
   B(z)= b0+b1*z+...+b_s z^s ( s=-1 =&gt; B(z)=0)
   D(z)= 1+d1*z+...+d_q*z^q  ( q=0 =&gt; D(z)=1)
   
    </pre>
    <p>
    for the method, see Eykhoff in trends and progress in system identification) page 96.
    with
    </p>
    <pre>
            z(t)=[y(t-1),..,y(t-r),u(t),...,
                  u(t-s),e(t-1),...,e(t-q)] 
    </pre>
    <p>
    and
    </p>
    <pre>
            coef= [-a1,..,-ar,b0,...,b_s,d1,...,d_q]'
            y(t)= coef'* z(t) + sig*e(t).
    </pre>
    <p>
    a sequential version of the AR estimation where e(t-i) is replaced 
    by an estimated value is used (RLLS). With q=0 this method is exactly
    a sequential version of armax</p>
    <h3>
      <font color="blue">Author</font>
    </h3>
    <p>J.-Ph.C; ;   </p>
  </body>
</html>
